biol 582

BIOL 582

Lecture Set 9

Factor Interactions

Factorial Models

BIOL 582 Factor interactions

• Many hypotheses in biological research are really interested in patterns of change:

• How do plant/animals traits change across environments? (e.g., phenotypic plasticity)

• How do traits change through development? (e.g., ontogenetics)

• Are patterns of variation constant across space or time? (e.g., spatial data)

• How do physiological responses change for organisms subjected to different treatments before and after a stimulus?


• Understanding the patterns of change becomes clear with factor interactions

• Interactions measure the joint effect of main effects A & B • Identifies whether response to A is dependent on level of B, or vice

versa• Are VERY common in biological research• Example: 2 species in 2 environments (Factors A & B), species 1 has

higher growth rate in moist environment, while species 2 has higher growth rate in dry environment. This would be identified as an interaction between species & environment

Growth rate

Wet Dry

Species 1

Species 2

Note: The study of trade-offs (reaction norms) in evolutionary ecology is based on the study of interactions


• Significant interactions identify a joint response of factors (response to Factor B depends on the level in Factor A)

• Interpreting interactions for two factors, each with two levels is straightforward (consider two species in two environments, for example)

From Collyer and Adams. (2007). Ecology. 88:683-692.

Growth rate

Wet Dry

Species 1

Species 2

01

23

4

V

E1 E2

01

23

4

V

E1 E2

01

23

4

V

E1 E2

01

23

4

V

E1 E2divergence/convergence

reversal of valueslarge effect/small effect

effect-no effect


• Consider what a true null hypothesis of no factor interaction might look like

• This is correct, right?

Growth rate

Wet Dry

Species 1

Species 2


• Consider what a true null hypothesis of no factor interaction might look like

• This is correct, right?

• So are these!

Growth rate

Wet Dry

Species 1

Species 2

Growth rate

Wet Dry

Species 1

Species 2

Wet Dry

Species 1

Species 2

Wet Dry

Species 1

Species 2

No species effect;

No environment effect

Small species effect;

Small environment effect

Small species effect;

Small environment effect

Large species effect;

No environment effect

Interactions mean paralleled patterns

of change


• Consider how a null hypothesis of no factor interaction would be rejected

• Important point: a significant interaction indicates differences in either the magnitude or direction of change (or both) between levels of one factor, among levels of the other factor

Growth rate

Wet Dry

Species 1

Species 2

Wet Dry

Species 1

Species 2

Wet Dry

Species 1

Species 2

One species changes; the other does not

Both species change in a similar direction; one at a greater rate

Both species change, but they change in opposite directions


• Consider how a null hypothesis of no factor interaction would be rejected when there are more than two levels of change (more possibilities exist!)

Growth rate

Wet Moist Dry

Species 1

Species 2

Growth rate

Wet Moist Dry

Species 1

Species 2

BIOL 582 Factorial Model Set-up

• First, consider this linear equation

• Which has the model

• That produces error

BIOL 582 Factorial Model Set-up

• Possible “sub-models” (reduced models) of the full model. They are shown here in terms of decreasing complexity

• Imagine that for every model, the SSE can be obtained easily (from residuals of predictions made by estimated model parameters).

• There are five sets of SSE from the four different models

• From model containing: both factors & interaction both factors only A factor only B factor only intercept only

• All models contain an intercept

BIOL 582 Factorial Model Hypotheses

Null Alternative Base Statistic

SSM = SST - SSE

SSA

SSB

SSAB

Note: When using Type I SS, the order of factor introduction can be important (see examples in R)

Note: One can use Types 1, II, or III SS. More on this in a moment.

BIOL 582 Factorial Model Uses and Assumptions

• There are MANY uses for factorial models in biological research

• Randomized Block designs: Subjects are randomized to treatments, within blocks. Blocks are experimental replicates. Block effects and interactions can be evaluated to consider extraneous sources of variation.

• Temporal considerations (time as a factor)• Spatial considerations (geography, altitude as a factor)• Sexual dimorphism (sex as a factor)• ETC.!

• Assumptions include• Normally distributed residuals (not data)• Homoscedasticity• Independent observations (i.e., sample sizes don’t contain multiple

measurements on the same subjects; different samples or treatments do not contain the same subjects)

• These are the assumptions of Linear Models!

BIOL 582 Factorial Model Evaluation

• Summary of ANOVA for two-factor factorial models• Type I (Sequential) – values in blue only necessary for F distribution-determination of P-values.

• Type III (Weighted)

• * This approach assumes that factor A is already in the model• k is the number of parameters (coefficients) needed for the effect

Source SS df MS F

A SSEμ - SSEA kA MSA = SSA/dfA MSA/MSE

B SSEA – SSEA,B * kB MSB = SSB/dfB MSB/MSE

AB SSEA,B - SSEA,B,AB kAkB MSAB = SSAB/dfAB MSAB/MSE

error SSEA,B,AB n – kA - kB- kAkB-1 MSE=SSEA,B,AB/dferror

Source SS df MS F

A SSEB,AB – SSEA,B,AB kA MSA = SSA/dfA MSA/MSE

B SSEA,AB – SSEA,B,AB kB MSB = SSB/dfB MSB/MSE

AB SSEA,B – SSEA,B,AB kAkB MSAB = SSAB/dfAB MSAB/MSE

error SSEA,B,AB n – kA - kB- kAkB-1 MSE=SSEAB/dferror


• Summary of ANOVA for two-factor factorial models• Type II (Partially Sequential)

• Type II SS does not seem remarkably different than type I SS, but the difference is more profound with complex models.

• For example, consider a 3-factor factorial model, which has effects A, B, C, AB, AC, BC, ABC

• For type III SS the reduced model for interactions removes only the interaction of interest; for type II SS, reduced models remove the factor or interaction plus any subsequent interactions

• Greater than 2-factor factorials are hard to evaluate, but it is worth considering the 3-factor factorial for the sake of SS types.

Source SS df MS F

A SSEB- SSEA,B kA MSA = SSA/dfA MSA/MSE

B SSEA – SSEA,B kB MSB = SSB/dfB MSB/MSE

AB SSEA,B - SSEA,B,AB kAkB MSAB = SSAB/dfAB MSAB/MSE

error SSEA,B,AB n – kA - kB- kAkB-1 MSE=SSEA,B,AB/dferror


• Summary of ANOVA for three-factor factorial models• Type III (Weighted) – F stat calculations removed for simplicity

Source SS

A SSE ,B,C,AB,AC,BC,ABC – SSEA,B,C,AB,AC,BC,ABC

B SSEA, ,C,AB,AC,BC,ABC – SSEA,B,C,AB,AC,BC,ABC

C SSEA,B, ,AB,AC,BC,ABC – SSEA,B,C,AB,AC,BC,ABC

AB SSEA,B,C, ,AC,BC,ABC – SSEA,B,C,AB,AC,BC,ABC

AC SSEA,B,C,AB, ,BC,ABC – SSEA,B,C,AB,AC,BC,ABC

BC SSEA,B,C,AB,AC, ,ABC – SSEA,B,C,AB,AC,BC,ABC

ABC SSEA,B,C,AB,AC,BC, – SSEA,B,C,AB,AC,BC,ABC

error SSEA,B,C,AB,AC,BC,ABC


• Summary of ANOVA for three-factor factorial models• Type II (Partially Sequential) – F stat calculations removed

Source SS

A SSE ,B,C, , ,BC, – SSEA,B,C, , ,BC,

B SSEA, ,C, ,AC, , – SSEA,B,C, ,AC, ,

C SSEA,B, ,AB, , , – SSEA,B,C,AB, , ,

AB SSEA,B,C, ,AC,BC, – SSEA,B,C,AB,AC,BC,

AC SSEA,B,C,AB, ,BC, – SSEA,B,C,AB,AC,BC,

BC SSEA,B,C,AB,AC, , – SSEA,B,C,AB,AC,BC,

ABC SSEA,B,C,AB,AC,BC, – SSEA,B,C,AB,AC,BC,ABC



• Summary of ANOVA for three-factor factorial models• Type I (Fully Sequential) – F stat calculations removed

• Multi-factor factorial models can get kind of crazy• There are various pros and cons to using different SS types – beyond our worries, for the most part• Just remember that ANOVA is nothing more than a comparison of errors of different models• Having clarity in realizing two different models that should be compared is ALL YOU NEED TO KNOW• Canned ANOVA in R or other software will try to make tables like above, but not every row is always needed• Biological intuition is a smart way to go• Creating your own table with “relevant” sources of variation is sufficient

Source SS

A SSEμ– SSEA

B SSEA – SSEA,B

C SSEA,B – SSEA,B,C

AB SSEA,B,C – SSEA,B,C,AB

AC SSEA,B,C,AB – SSEA,B,C,AB,AC

BC SSEA,B,C,AB,AC – SSEA,B,C,AB,AC,BC

ABC SSEA,B,C,AB,AC,BC – SSEA,B,C,AB,AC,BC,ABC


BIOL 582 Factorial Model ANOVA Example

• Example from pupfish-parasite data in R> log.grubs<-log(GRUBS+1)> > # TYPE I SS, two ways> > lm.pop.sex<-lm(log.grubs~POPULATION*SEX)> lm.sex.pop<-lm(log.grubs~SEX*POPULATION)> > anova(lm.pop.sex)

Analysis of Variance Table

Response: log.grubs Df Sum Sq Mean Sq F value Pr(>F) POPULATION 1 0.088 0.0879 0.0553 0.814652 SEX 1 16.643 16.6425 10.4601 0.001658 **POPULATION:SEX 1 6.606 6.6055 4.1517 0.044261 * Residuals 99 157.513 1.5910 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > anova(lm.sex.pop)


Response: log.grubs Df Sum Sq Mean Sq F value Pr(>F) SEX 1 15.554 15.5543 9.7762 0.002321 **POPULATION 1 1.176 1.1762 0.7392 0.391980 SEX:POPULATION 1 6.606 6.6055 4.1517 0.044261 * Residuals 99 157.513 1.5910 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 >


• Example from pupfish-parasite data in R> qqnorm(resid(lm.pop.sex))

> shapiro.test(resid(lm.pop.sex))

Shapiro-Wilk normality test

data: resid(lm.pop.sex)

W = 0.9869, p-value = 0.4061

> qqnorm(resid(lm.sex.pop))

> shapiro.test(resid(lm.sex.pop))

Shapiro-Wilk normality test

data: resid(lm.sex.pop)

W = 0.9869, p-value = 0.4061


• Example from pupfish-parasite data in R> par(mfrow=c(1,2))

> plot(predict(lm.pop.sex),(resid(lm.pop.sex)-mean(resid(lm.pop.sex))/sd(resid(lm.pop.sex))),xlab="Predicted Values",ylab="Standardized Residuals",main="lm.pop.sex")

> plot(predict(lm.sex.pop),(resid(lm.sex.pop)-mean(resid(lm.sex.pop))/sd(resid(lm.sex.pop))),xlab="Predicted Values",ylab="Standardized Residuals",main="lm.sex.pop")


• Example from pupfish-parasite data in R> log.grubs<-log(GRUBS+1)> > # TYPE I SS, two ways> > lm.pop.sex<-lm(log.grubs~POPULATION*SEX)> lm.sex.pop<-lm(log.grubs~SEX*POPULATION)> > anova(lm.pop.sex)


Response: log.grubs Df Sum Sq Mean Sq F value Pr(>F) POPULATION 1 0.088 0.0879 0.0553 0.814652 SEX 1 16.643 16.6425 10.4601 0.001658 **POPULATION:SEX 1 6.606 6.6055 4.1517 0.044261 * Residuals 99 157.513 1.5910 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > anova(lm.sex.pop)


Response: log.grubs Df Sum Sq Mean Sq F value Pr(>F) SEX 1 15.554 15.5543 9.7762 0.002321 **POPULATION 1 1.176 1.1762 0.7392 0.391980 SEX:POPULATION 1 6.606 6.6055 4.1517 0.044261 * Residuals 99 157.513 1.5910 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 >


• Example from pupfish-parasite data in R> # TYPE III SS, two ways> > lm.pop.sex<-lm(log.grubs~POPULATION*SEX)> lm.sex.pop<-lm(log.grubs~SEX*POPULATION)> > drop1(lm.pop.sex,test="F")

Single term deletions

Model:log.grubs ~ POPULATION * SEX Df Sum of Sq RSS AIC F value Pr(F) <none> 157.51 51.752 POPULATION:SEX 1 6.6055 164.12 53.984 4.1517 0.04426 *---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > drop1(lm.sex.pop,test="F")


Model:log.grubs ~ SEX * POPULATION Df Sum of Sq RSS AIC F value Pr(F) <none> 157.51 51.752 SEX:POPULATION 1 6.6055 164.12 53.984 4.1517 0.04426 *---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 >


• Example from pupfish-parasite data in> # R will not "go all the way" so one can try the following> > lm.pop.by.sex<-lm(log.grubs~POPULATION*SEX)> lm.pop.and.sex<-lm(log.grubs~POPULATION+SEX)> > drop1(lm.pop.by.sex,test="F")


Model:log.grubs ~ POPULATION * SEX Df Sum of Sq RSS AIC F value Pr(F) <none> 157.51 51.752 POPULATION:SEX 1 6.6055 164.12 53.984 4.1517 0.04426 *---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > drop1(lm.pop.and.sex,test="F")


Model:log.grubs ~ POPULATION + SEX Df Sum of Sq RSS AIC F value Pr(F) <none> 164.12 53.984 POPULATION 1 1.1762 165.29 52.719 0.7167 0.399264 SEX 1 16.6425 180.76 61.932 10.1405 0.001934 **---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 >

WRONG

WRONG This is an admixture of SS types


• Example from pupfish-parasite data in R> # The following requires using the 'car' (companion to applied regression) package> # you might have to install it.> # Not needed for type 1> > library(car)> > anova(lm.pop.sex)


Response: log.grubs Df Sum Sq Mean Sq F value Pr(>F) POPULATION 1 0.088 0.0879 0.0553 0.814652 SEX 1 16.643 16.6425 10.4601 0.001658 **POPULATION:SEX 1 6.606 6.6055 4.1517 0.044261 * Residuals 99 157.513 1.5910 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> anova(lm.sex.pop)


Response: log.grubs Df Sum Sq Mean Sq F value Pr(>F) SEX 1 15.554 15.5543 9.7762 0.002321 **POPULATION 1 1.176 1.1762 0.7392 0.391980 SEX:POPULATION 1 6.606 6.6055 4.1517 0.044261 * Residuals 99 157.513 1.5910 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


• Example from pupfish-parasite data in R> # The following requires using the 'car' (companion to applied regression) package> # you might have to install it.> # Not needed for type 1> > Anova(lm.pop.sex, type="II")

Anova Table (Type II tests)

Response: log.grubs Sum Sq Df F value Pr(>F) POPULATION 1.176 1 0.7392 0.391980 SEX 16.643 1 10.4601 0.001658 **POPULATION:SEX 6.606 1 4.1517 0.044261 * Residuals 157.513 99 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> Anova(lm.sex.pop, type="II")

Anova Table (Type II tests)

Response: log.grubs Sum Sq Df F value Pr(>F) SEX 16.643 1 10.4601 0.001658 **POPULATION 1.176 1 0.7392 0.391980 SEX:POPULATION 6.606 1 4.1517 0.044261 * Residuals 157.513 99 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


• Example from pupfish-parasite data in R> # The following requires using the 'car' (companion to applied regression) package> # you might have to install it.> # Not needed for type 1> > Anova(lm.pop.sex, type="III")

Anova Table (Type III tests)

Response: log.grubs Sum Sq Df F value Pr(>F) (Intercept) 99.054 1 62.2572 4.119e-12 ***POPULATION 0.632 1 0.3973 0.52995 SEX 1.307 1 0.8214 0.36696 POPULATION:SEX 6.606 1 4.1517 0.04426 * Residuals 157.513 99 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> Anova(lm.sex.pop, type="III")

Anova Table (Type III tests)

Response: log.grubs Sum Sq Df F value Pr(>F) (Intercept) 99.054 1 62.2572 4.119e-12 ***SEX 1.307 1 0.8214 0.36696 POPULATION 0.632 1 0.3973 0.52995 SEX:POPULATION 6.606 1 4.1517 0.04426 * Residuals 157.513 99 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


• General Rule of Thumb• If the interaction is significant, do not evaluate “main” effects; just evaluate

the interaction• If the interaction is not significant, evaluate the main effects (ignore the

interaction or remove it from the model)

> interaction.plot(POPULATION,SEX,log.grubs)


• General Rule of Thumb• If the interaction is significant, do not evaluate “main” effects; just evaluate

the interaction• If the interaction is not significant, evaluate the main effects (ignore the

interaction or remove it from the model)

> library(gplots) # requires that gplots package is installed

> group<-factor(paste(SEX,POPULATION,SEP="."))

> plotmeans(log.grubs~group)

BIOL 582 Multiple Comparisons

• SS Type is not important. • Multiple comparison tests like Tukey’s HSD use the SSE of the full

model to calculate standard error.• Example from pupfish-parasite data in R

> sex<-factor(SEX); pop<-factor(POPULATION)> > TukeyHSD(aov(log.grubs~pop*sex)) Tukey multiple comparisons of means 95% family-wise confidence level

Fit: aov(formula = log.grubs ~ pop * sex)

$pop diff lwr upr p adj2-1 0.05856328 -0.4358007 0.5529273 0.8146523

$sex diff lwr upr p adjM-F 0.805493 0.3016897 1.309296 0.0020137

$`pop:sex` diff lwr upr p adj2:F-1:F -0.2072937 -1.0667296 0.6521422 0.92205901:M-1:F 0.3153772 -0.6361571 1.2669114 0.82227642:M-1:F 1.1630006 0.1180954 2.2079059 0.02287311:M-2:F 0.5226708 -0.3367651 1.3821067 0.38943272:M-2:F 1.3702943 0.4085045 2.3320841 0.00183192:M-1:M 0.8476235 -0.1972817 1.8925287 0.1539578

BIOL 582 Final Remarks

• Factorial ANOVA is essentially one-way ANOVA broken into subcomponents of variation.

• Consider this example

> lm.sex.pop<-lm(log.grubs~sex*pop)> group<-factor(paste(SEX,POPULATION,SEP="."))> lm.group<-lm(log.grubs~group)> > summary(lm.sex.pop)

Call:lm(formula = log.grubs ~ sex * pop)

Residuals: Min 1Q Median 3Q Max -2.8590 -0.8546 0.1149 0.7944 2.8303

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.1649 0.2575 16.176 <2e-16 ***sexM 0.3154 0.3641 0.866 0.3885 pop2 -0.2073 0.3289 -0.630 0.5300 sexM:pop2 1.0549 0.5177 2.038 0.0443 * ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.261 on 99 degrees of freedomMultiple R-squared: 0.129, Adjusted R-squared: 0.1026 F-statistic: 4.889 on 3 and 99 DF, p-value: 0.003273




> lm.sex.pop<-lm(log.grubs~sex*pop)> group<-factor(paste(SEX,POPULATION,SEP="."))> lm.group<-lm(log.grubs~group)> > summary(lm.group)

Call:lm(formula = log.grubs ~ group)

Residuals: Min 1Q Median 3Q Max -2.8590 -0.8546 0.1149 0.7944 2.8303

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.1649 0.2575 16.176 < 2e-16 ***groupF 2 . -0.2073 0.3289 -0.630 0.52995 groupM 1 . 0.3154 0.3641 0.866 0.38852 groupM 2 . 1.1630 0.3999 2.909 0.00448 ** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.261 on 99 degrees of freedomMultiple R-squared: 0.129, Adjusted R-squared: 0.1026 F-statistic: 4.889 on 3 and 99 DF, p-value: 0.003273




> TukeyHSD(aov(log.grubs~sex*pop)) Tukey multiple comparisons of means 95% family-wise confidence level

Fit: aov(formula = log.grubs ~ sex * pop)

$sex diff lwr upr p adjM-F 0.7938817 0.2900784 1.297685 0.0023212

$pop diff lwr upr p adj2-1 0.2101225 -0.2842415 0.7044865 0.4010581

$`sex:pop` diff lwr upr p adjM:1-F:1 0.3153772 -0.6361571 1.2669114 0.8222764F:2-F:1 -0.2072937 -1.0667296 0.6521422 0.9220590M:2-F:1 1.1630006 0.1180954 2.2079059 0.0228731F:2-M:1 -0.5226708 -1.3821067 0.3367651 0.3894327M:2-M:1 0.8476235 -0.1972817 1.8925287 0.1539578M:2-F:2 1.3702943 0.4085045 2.3320841 0.0018319

> TukeyHSD(aov(log.grubs~group)) Tukey multiple comparisons of means 95% family-wise confidence level

Fit: aov(formula = log.grubs ~ group)

$group diff lwr upr p adjF.2-F.1 -0.2072937 -1.0667296 0.6521422 0.9220590M.1-F.1 0.3153772 -0.6361571 1.2669114 0.8222764M.2-F.1 1.1630006 0.1180954 2.2079059 0.0228731M.1-F.2 0.5226708 -0.3367651 1.3821067 0.3894327M.2-F.2 1.3702943 0.4085045 2.3320841 0.0018319M.2-M.1 0.8476235 -0.1972817 1.8925287 0.1539578


• Next time using R, try this

• or

• It is super cool!

> plot(lm.sex.pop)

> par(mfcol=c(2,2))> plot(lm.sex.pop)

biol 582

Documents

factor b

b factor

factor interactionsconsider

species environmentnote

notboth species

factorone species

factors interaction

direction of change